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There! Should do for now. Paidgenius 20:24, 21 November 2006 (UTC)
Well, everyone makes mistakes. That's cool though. Paidgenius 20:12, 8 December 2006 (UTC)
I am unclear why my last edit should be removed. I added a sentence to say that B would shoot at C rather than A. Doctormatt removed it, saying it was unsupported, and the question is A's strategy, rather than B's. I also think that the figures given for chances of survival are incorrect.
C's best choice is to shoot B if both B and C survive, because B has the best chance of killing him. B's best choice is to shoot C, as C will kill him if he shoots at him. If A kills B, then C will kill A. If A kills C, B has a 2/3 chance of killing A. So A shoots to miss.
Then B shoots at C. In 6/9 examples, B will kill C. It is then A's turn again: in 2/9 examples, A will kill B and survive. In 3/9 examples, B will miss C. C will then kill B. A will shoot C, and in 1/9 examples A will kill C and survive. In 2/9 examples, A will miss C, and C will then kill A. In 4/9 examples, B will shoot at A having killed C, and have a 2/3 chance of killing A. Abigailgem 16:59, 29 May 2007 (UTC)
Having had a look at some of the previous revisions, I still would prefer a verbal as well as mathematical explanation of the solution. I am placing that here rather than making such an extreme revision. Does anyone have a comment?
Given a choice between shooting at A or B, C should shoot B, who has the best chance of killing C. Therefore, given a choice of shooting A or C, B should choose to shoot at C.
If A kills B, A will then be killed by C. If A kills C, he faces B and it is B's shot. So A shoots to miss.
In 1/3 of cases, B will miss, then C will kill B and A then has one shot at C, a one third chance of success.
In 2/3 of cases, B will kill C, so that A faces B and it is A's shot.
I do not propose to alter the calculations. Abigailgem 14:44, 28 July 2007 (UTC)
Indeed. But, if A shoots with the intention of hitting B, in one third of cases (according to the way the problem is drafted) A will kill B. This is an outcome against A's interests, as C, the perfect shot, will then kill A. This is part of the rules of the puzzle, as it is set. There is no need to state it in the solution. The point is that A does not want to kill B or C with his first shot, as that will harm his interests.
I consider changing the example would not be an improvement of the article. However, adding an additional example which was non-intuitive could be an improvement, giving other aspects of the problem. Abigailgem 15:31, 5 August 2007 (UTC)
An editor removed the example of a theoretical truel, claiming that it is "blatant OR". My feeling is that this is not OR, as any undergraduate probability student can verify this example. It is not "research": it is merely a not completely trivial (but not difficult) computation. In mathematics articles there are numerous such examples of the results of computations. In terms of verifiability, anyone with a modicum of probability knowledge can verify this, without any other references, so I don't think any citations are needed for this. I'd be happy to hear what other think. I'd also be happy to typset the calculations, and perhaps put them on a separate "proof" page. If people really feel it's OR, I'll find a published example to copy. Doctormatt 22:57, 4 November 2007 (UTC)
In a book by Martin Gardner, there is a solved problem about a truel. I suspect this is the "original" truel problem, others being variations of this one. From memory: Cowboy Smith is a 100% sure shot, Brown 80% and Jones 50%. They shoot sequentially, the order set by a lottery from the start. They have infinitely many bullets, they can deliberately miss, and they keep on shooting till only one is alive. Last man standing wins the girl they are fighting about. As long as S and B are both alive, they will shoot at each others only, and J should deliberately miss - because if he shoots one of them, it will be the other one's turn, and he will die at the next shot with 80% or 100% certainty. If he waits till S or B has finished off the other one, it will be his turn next, and J has a 50% chance of winning at the next shot. Completing the probability tree (including a geometric series),
This contradicts the statement presently in the article that it is the best strategy to shoot at the best shooter - not so for Jones!-- Noe ( talk) 15:40, 24 November 2008 (UTC)
I propose merging Three way duel (puzzle) into this article ( Truel), as they seem to deal with essentially the same topic. I chose to propose a merge into this article because this article seems to have a longer history and more activity. However, one could make an argument for a merge in the other direction.-- GregRM ( talk) 04:37, 23 December 2008 (UTC)
OK, the following ought to be merged nicely into the article - I'll try to get back to this if not someone beats me to it ;-) -- Nø ( talk) 14:25, 23 September 2010 (UTC)
The three way duel is a logic puzzle proposed by Martin Gardner. [1]
Three men are in a pistol duel. Each man will shoot in turn. The three men are identified as A, B, and C. A is a poor shot, and hits only 50% of the time. B is an expert marksman, and always hits. C is a moderate shot, and hits 80% of the time. (In some variants of the problem, A's probability is 25% and C's is 50%; in practice this makes little difference as long as A's is lower than C's.)
The exact order in which the three men will take turns shooting is variable (although in some variants it is stated as being A, B, C).
As the problem is based on classical dueling it is assumed that a hit target is "killed", but this is not necessary and some variants of the problem alter the theme to a game where targets are unharmed, but required to withdraw, when hit.
The question is, what is the best strategy for A to follow to win the duel?
The most common solution [2] is that A should deliberately fire into the ground until one of the other two men is dead, then shoot at him.
The reasoning for this is as follows: B and C are greater threats to each other than A is to either of them, and thus rationally should target each other first. If B fires first, he will certainly kill C; if C fires first, he may kill B, and if he does not, B will certainly kill him. In the first "round" of the duel one of these two interactions will occur between B and C and it is not in A's interest to disrupt it. If A kills one of the men, the other man will target him, probably killing him; if A fires but does not kill either man, he makes no difference. After either of these interactions completes, it will become A's turn with A never having been targeted and having a chance to kill the one remaining man and win the duel, and this is the best possible position for him.
This solution is considered by some observers to expose a paradox in reasoning. They argue that, if A is permitted to fire into the ground, then B and C could do so too. Based on that, if survival is a duelist's only priority, the best strategy is for all three men to fire into the ground until the ammo runs out and then walk away. Intuitively and emotionally we tend to believe that if A dies by the end of the duel, it makes no difference how many people he killed before he died; but using this heuristic to judge duel strategies may lead to this strategy being considered optimal. If "winning the duel" has additional requirements other than survival (eg, killing the opponents), A's strategy above might no longer be considered the best because it fails to meet those other requirements. Although since B is going to be the first one to get shot anyway, he might as well shoot someone (C if he is still there), therefore C should shoot him, so this strategy does work only for A. If it is allowed, B could also shoot himself non-fatally, so as to make sure he was not the first shot, but whether or not this would actually make sense would depend on how much of a difference this makes in B's ability to aim.
You can't be serious about that math equation. I know that Wikipedia is full (F-U-L-L) of useful information, but sometimes, one could live without such nonsense. An explanation of the topic, which, depending on the cultural position and popularity of the topic itself, etc., etc., etc., can vary in depth; the history of the topic; and other information about it should be all that is needed. Obviously, most people don't want a math equation about a word that is similar to the word "duel". — Preceding unsigned comment added by Kaabii123 ( talk • contribs) 21:10, 3 June 2012 (UTC)
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There! Should do for now. Paidgenius 20:24, 21 November 2006 (UTC)
Well, everyone makes mistakes. That's cool though. Paidgenius 20:12, 8 December 2006 (UTC)
I am unclear why my last edit should be removed. I added a sentence to say that B would shoot at C rather than A. Doctormatt removed it, saying it was unsupported, and the question is A's strategy, rather than B's. I also think that the figures given for chances of survival are incorrect.
C's best choice is to shoot B if both B and C survive, because B has the best chance of killing him. B's best choice is to shoot C, as C will kill him if he shoots at him. If A kills B, then C will kill A. If A kills C, B has a 2/3 chance of killing A. So A shoots to miss.
Then B shoots at C. In 6/9 examples, B will kill C. It is then A's turn again: in 2/9 examples, A will kill B and survive. In 3/9 examples, B will miss C. C will then kill B. A will shoot C, and in 1/9 examples A will kill C and survive. In 2/9 examples, A will miss C, and C will then kill A. In 4/9 examples, B will shoot at A having killed C, and have a 2/3 chance of killing A. Abigailgem 16:59, 29 May 2007 (UTC)
Having had a look at some of the previous revisions, I still would prefer a verbal as well as mathematical explanation of the solution. I am placing that here rather than making such an extreme revision. Does anyone have a comment?
Given a choice between shooting at A or B, C should shoot B, who has the best chance of killing C. Therefore, given a choice of shooting A or C, B should choose to shoot at C.
If A kills B, A will then be killed by C. If A kills C, he faces B and it is B's shot. So A shoots to miss.
In 1/3 of cases, B will miss, then C will kill B and A then has one shot at C, a one third chance of success.
In 2/3 of cases, B will kill C, so that A faces B and it is A's shot.
I do not propose to alter the calculations. Abigailgem 14:44, 28 July 2007 (UTC)
Indeed. But, if A shoots with the intention of hitting B, in one third of cases (according to the way the problem is drafted) A will kill B. This is an outcome against A's interests, as C, the perfect shot, will then kill A. This is part of the rules of the puzzle, as it is set. There is no need to state it in the solution. The point is that A does not want to kill B or C with his first shot, as that will harm his interests.
I consider changing the example would not be an improvement of the article. However, adding an additional example which was non-intuitive could be an improvement, giving other aspects of the problem. Abigailgem 15:31, 5 August 2007 (UTC)
An editor removed the example of a theoretical truel, claiming that it is "blatant OR". My feeling is that this is not OR, as any undergraduate probability student can verify this example. It is not "research": it is merely a not completely trivial (but not difficult) computation. In mathematics articles there are numerous such examples of the results of computations. In terms of verifiability, anyone with a modicum of probability knowledge can verify this, without any other references, so I don't think any citations are needed for this. I'd be happy to hear what other think. I'd also be happy to typset the calculations, and perhaps put them on a separate "proof" page. If people really feel it's OR, I'll find a published example to copy. Doctormatt 22:57, 4 November 2007 (UTC)
In a book by Martin Gardner, there is a solved problem about a truel. I suspect this is the "original" truel problem, others being variations of this one. From memory: Cowboy Smith is a 100% sure shot, Brown 80% and Jones 50%. They shoot sequentially, the order set by a lottery from the start. They have infinitely many bullets, they can deliberately miss, and they keep on shooting till only one is alive. Last man standing wins the girl they are fighting about. As long as S and B are both alive, they will shoot at each others only, and J should deliberately miss - because if he shoots one of them, it will be the other one's turn, and he will die at the next shot with 80% or 100% certainty. If he waits till S or B has finished off the other one, it will be his turn next, and J has a 50% chance of winning at the next shot. Completing the probability tree (including a geometric series),
This contradicts the statement presently in the article that it is the best strategy to shoot at the best shooter - not so for Jones!-- Noe ( talk) 15:40, 24 November 2008 (UTC)
I propose merging Three way duel (puzzle) into this article ( Truel), as they seem to deal with essentially the same topic. I chose to propose a merge into this article because this article seems to have a longer history and more activity. However, one could make an argument for a merge in the other direction.-- GregRM ( talk) 04:37, 23 December 2008 (UTC)
OK, the following ought to be merged nicely into the article - I'll try to get back to this if not someone beats me to it ;-) -- Nø ( talk) 14:25, 23 September 2010 (UTC)
The three way duel is a logic puzzle proposed by Martin Gardner. [1]
Three men are in a pistol duel. Each man will shoot in turn. The three men are identified as A, B, and C. A is a poor shot, and hits only 50% of the time. B is an expert marksman, and always hits. C is a moderate shot, and hits 80% of the time. (In some variants of the problem, A's probability is 25% and C's is 50%; in practice this makes little difference as long as A's is lower than C's.)
The exact order in which the three men will take turns shooting is variable (although in some variants it is stated as being A, B, C).
As the problem is based on classical dueling it is assumed that a hit target is "killed", but this is not necessary and some variants of the problem alter the theme to a game where targets are unharmed, but required to withdraw, when hit.
The question is, what is the best strategy for A to follow to win the duel?
The most common solution [2] is that A should deliberately fire into the ground until one of the other two men is dead, then shoot at him.
The reasoning for this is as follows: B and C are greater threats to each other than A is to either of them, and thus rationally should target each other first. If B fires first, he will certainly kill C; if C fires first, he may kill B, and if he does not, B will certainly kill him. In the first "round" of the duel one of these two interactions will occur between B and C and it is not in A's interest to disrupt it. If A kills one of the men, the other man will target him, probably killing him; if A fires but does not kill either man, he makes no difference. After either of these interactions completes, it will become A's turn with A never having been targeted and having a chance to kill the one remaining man and win the duel, and this is the best possible position for him.
This solution is considered by some observers to expose a paradox in reasoning. They argue that, if A is permitted to fire into the ground, then B and C could do so too. Based on that, if survival is a duelist's only priority, the best strategy is for all three men to fire into the ground until the ammo runs out and then walk away. Intuitively and emotionally we tend to believe that if A dies by the end of the duel, it makes no difference how many people he killed before he died; but using this heuristic to judge duel strategies may lead to this strategy being considered optimal. If "winning the duel" has additional requirements other than survival (eg, killing the opponents), A's strategy above might no longer be considered the best because it fails to meet those other requirements. Although since B is going to be the first one to get shot anyway, he might as well shoot someone (C if he is still there), therefore C should shoot him, so this strategy does work only for A. If it is allowed, B could also shoot himself non-fatally, so as to make sure he was not the first shot, but whether or not this would actually make sense would depend on how much of a difference this makes in B's ability to aim.
You can't be serious about that math equation. I know that Wikipedia is full (F-U-L-L) of useful information, but sometimes, one could live without such nonsense. An explanation of the topic, which, depending on the cultural position and popularity of the topic itself, etc., etc., etc., can vary in depth; the history of the topic; and other information about it should be all that is needed. Obviously, most people don't want a math equation about a word that is similar to the word "duel". — Preceding unsigned comment added by Kaabii123 ( talk • contribs) 21:10, 3 June 2012 (UTC)